Volume 17, issue 5 (2013)

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A geometric transition from hyperbolic to anti-de Sitter geometry

Jeffrey Danciger

Geometry & Topology 17 (2013) 3077–3134
Abstract

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti-de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the “other side” of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these methods in the case when the manifold is the unit tangent bundle of the $\left(2,m,m\right)$ triangle orbifold for $m\ge 5$.

Keywords
geometric transition, hyperbolic, AdS, cone manifold, tachyon, projective structure, transitional geometry, half-pipe geometry
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 53C15, 53B30, 20H10, 53C30